Atomic spectroscopy (part I), 2017 Uwe Burghaus, Fargo, ND, USA

Transcription

1 Atomic spectroscopy (part I), 2017 Uwe Burghaus, Fargo, ND, USA

2 Last class: group theory Symmetry operations σ reflections C rotations S rotation-reflections I inversion E identity

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4 Spectra of Atoms and Molecules, 3 rd Ed., Peter F. Bernath, Oxford University Press, chapter 5 Physics of Atoms and Molecules, B.H. Bransden, C.J. Joachain, Wiley, chapters 3-9 Foundations of Spectroscopy, S. Duckett, B. Gilbert, Oxford Chemistry Primers, Vol. 78 chapter 4 Molecular Spectroscopy, J.M. Brown, Oxford Chemistry Primers, Vol. 55 chapter 7

5 Jespersen, Hyslop, Chemistry the molecular nature of matter chapter 7.2; 7.3, page 338 (7 th Ed.) or R. Chang, Physical Chemistry for the Biosciences chapter 11.4, page 407- or P. Atkins / J. Paula, Physical chemistry for the life sciences chapter 9.7, page 338 (2 nd Ed.) or P. Atkins / J. Paula, Physical Chemistry chapter 11.1d, page 298 (7 th Ed.) or I.N. Levine, Physical chemistry chapter 18.3, page 602 (5 th Ed.) or T. Engel / P. Reid, Physical chemistry, chapter 12.7 (1 st Ed.) I PChem

6 Atomic spectroscopy Absorption spectroscopy Bohr model QM of H atom review Later in this class: Molecular spectroscopy, 2017 Uwe Burghaus, Fargo, ND, USA

7 Today Bohr Model

8 Niels Henrik David Bohr ( ) Danish physicist Nobel Prize in Physics in During 2 nd world war: UK USA Manhatten project in LosAlamos Also his son got a Nobel price. Similar case: Marie Curie & her daughter Who rejected a Nobel prize?

9 hydrogen (from the Greek ὑδρο- hydro meaning "water" and -γενής genes meaning "creator ): water is produced when hydrogen is burned Fe + H 2 O FeO + H 2 Robert Boyle apparently the 1 st who hade H 2 in 1671 Antoine-Laurent de Lavoisier named the element hydrogen in H 2 (g) + O 2 (g) 2 H 2 O(l)

10 Emission/Absorption spectra of atomic gases glowing gas sample Emission spectra of atoms bright line spectra PChem Quantum mechanics H He lamp Absorption spectra of atoms dark line spectra H Lyman series H Balmer series non-glowing gas sample H α H β H γ H οο

11 Emission/Absorption spectra of atomic gases PChem Quantum mechanics Experimental results Line spectra opposed to continuous blackbody radiation Different atoms different spectra Complex pattern of lines Series of lines Lines converge into continuum at one end

12 Emission (/Absorption) spectroscopy PChem Quantum mechanics light source slit lens diffraction gratings lens detector What is that, emission or absorption spectroscopy?

13 Light = electromagnetic wave PChem Quantum mechanics 1D figure Snapshot time = t 0 E(r,t) = E0cos(kr -ωt + φ) B(r,t) = B cos(kr -ωt + φ) electric field amplitude of electric E E k vector 0 field vector r coordinates ω angular t frequency φ phase lag B B 0 magnetic field vector 0 wave vector time amplitude of magnetic field vector Maxwell According to Maxwell's classical theory of light, electromagnetic waves are produced by accelerated electric charges. A charge oscillating at a frequency ν will emit radiation at that frequency.

14 Classical physics & spectroscopy PChem Quantum mechanics The spectroscopic results from Balmer, Lyman, etc. were inconsistent with classical physics. According to Maxwell's classical theory of light, electromagnetic waves are produced by accelerated electric charges. A charge oscillating at a frequency ν will emit radiation at that frequency. Bohr s postulates 1) Stationary states, E 1, E 2, E 3, Quantum mechanics Classical physics

15 Emission/Absorption spectra PChem Quantum mechanics Bohr s postulates 1) Stationary states, E 1, E 2, E 3, 2) No radiation on stationary states 3) E=E n -E n+1 = hν (transitions) 4) F(Coulomb) = F(Centrifugal) 5) Quantization of angular momentum l = n h/(2pi); n = 1,2,3, ν = 1/ λ = E E 4 2 e m0z 1 1 = ( ) π ε c m n n m Energy R (Rhydberg constant) m 0 : electron mass for speed = zero c: speed of light m, n: quantum # ε: dielectric constant Z: charge number λ: wavelength 1/λ: wave number e: electron charge h: Planck s const. h_bar = h/2π

16 Absorption / Emission of a photon Quantum mechanics E 2 E 2 photon E = hν E = hν photon E 1 E 1 absorption emission Bohr s postulates 1) Stationary states, E 1, E 2, E 3, 2) No radiation on stationary states 3) E=E n -E n+1 = hν (transitions).

17 Emission/Absorption spectra PChem Quantum mechanics ν = 1/ λ = E E 4 2 e m0z 1 1 = ( ) π ε c m n n m 0 Energy Energy 3 2 ν 1 n 2 R (Rhydberg constant) 0 1 m 0 : electron mass for speed = zero c: speed of light m, n: quantum # ε: dielectric constant Z: charge number λ: wavelength 1/λ: wave number e: electron charge h: Planck s const. h_bar = h/2π

18 Absorption spectra of atoms lamp non-glowing gas sample dark line spectra Classical limit n large (classical mechanics) ν = 1/ λ = E E 4 2 e m0z 1 1 = ( ) π ε c m n n m Electronic states of atoms n small: QM n large: classical mechanics Chang figure 11.12

19 Classical limit PChem Quantum mechanics QM classical mechanic n large h 0 Why do we know this? Spectroscopy, Bohr model Blackbody radiation Planck's equation Rayleigh s equation

20 Bohr s derivation PChem Quantum mechanics ν = 1/ λ = E E 4 2 e m0z 1 1 = ( ) π ε c m n n m Most books don t include the historic derivation, because it s simple but lengthy algebra, & it is not really correct centripetal force = coulomb force (4 th Bohr postulate) Quantization of angular momentum (5 th Bohr postulate) m 0 : electron mass for speed = zero c: speed of light m, n: quantum # ε: dielectric constant Z: charge number λ: wavelength 1/λ: wave number e: electron charge h: Planck s const. h_bar = h/2π k e Coulomb constant

21 Standing waves & stationary states in quantum mechanics PChem Quantum mechanics λ/2 2x λ/2 3x λ/2 Bohr s postulates 1) Stationary states, E 1, E 2, E 3, λ electron = h p Quantization of angular momentum (5 th Bohr postulate) wave fits in orbit wave does NOT fit in orbit

22 Important & well known numbers PChem Quantum mechanics How large is an atom? Bohr radius a 0 Ground state energy of H atom m 0 : electron mass for speed = zero c: speed of light m, n: quantum # ε: dielectric constant Z: charge number λ: wavelength 1/λ: wave number e: electron charge h: Planck s const. h_bar = h/2π k e Coulomb constant

23 More modern version of this looks like this lamp Absorption spectra of atoms dark line spectra diffraction grating light source mirror reference cell detector non-glowing gas sample 1.0 transparent mirror absorption (a. u.) Anthracene Absorption spectra of molecules mirror mirror sample cell detector wavelength (nm) Anthracene data from NDSU Pchem lab class 2006, U. Burghaus

24 Vibrations of small molecules - UVvis Absorption spectra of small molecules vibrations 1.0 electronic states absorption (a. u.) Anthracene wavelength (nm) Excitation spectra excitation wavelength scanned emission wavelength fixed Anthracene data from NDSU Pchem lab class 2006, U. Burghaus

25 Vibrations and rotations of small molecules - FTIR energy Electronic excitations vibrations rotations of molecules. PChem364 quantum mechanics excited state rotations distance j=3 j=2 j=1 j=0 v=3 v=2 electronic energy levels ground state v=0 v=1 distance electronic energy level vibrations

26 HCl/DCl IR data from NDSU Pchem lab class 2008 Standard example rot & vis spectra of HCl, DCl Vibration-rotation EXAMPLE HCl/DCl inensity (a.u.) 70 P branch R branch isotope splitting wave number in 1/cm inensity (a.u.) wave number in 1/cm v = 1 v = 0 j = 3 j = 2 j = 1 j = 0 j = 3 j = 2 j = 1 j = 0

27 31 Electronic structure of carbon nanotubes E 2 E 1 photon absorption E = hν metallic carbon nanotubes semiconducting carbon nanotubes energy conduction band energy conduction band E F E 11 E 11 E 22 E F valence band valence band Density of states Density of states

28 Absorption spectroscopy Energy 3 2 absorption (a. u.) 0.6 Anthracene wavelength (nm) energy conduction band vibrations electronic states E F 1 valence band Density of states H

29 So far this was a review of non chemistry major undergraduate knowledge

30 Atomic spectroscopy Absorption spectroscopy Bohr model QM of H atom review Later in this class: Molecular spectroscopy, 2017 Uwe Burghaus, Fargo, ND, USA

31 Survey of the quantum mechanics of the H atom Goal for the rest of this class Understand structure of the solution No mathematical details today Interaction potential coulomb potential Hamilton operator for H atom Definition of the problem Strategy spherical coordinates Idea of the solution Separation of variables Solution three equations quantum numbers

32 Spherical Coordinates PChem Quantum mechanics z (r,θ,φ) (x,y,z) z x φ Θ r y Θ r y (x,y,z) dv = dxdydz x (r,θ,φ) dv = r 2 dr sin(θ) dθ dφ

33 PChem Quantum mechanics H atom mathematical structure of the solution ( ) H x y z Ze r H E = + + = µ πε 0 ψ ψ ; Idea: use spherical coordinates, separate variables 3-separte differential equations ψ ψ πε φ ψ ψ ψ E r e r r r r r r m = Θ + Θ Θ Θ Θ )) ( sin 1 ) (sin sin 1 ) ( 1 [ 2 ) ( ) ( ) (,,,, φ θ ψ m m l l n m l n r R Φ Θ =

34 Schrödinger equation for the H atom in polar coordinates 2 Eq. (20.2) in Engel/Reid Compare this with 3D rigid rotor (chapter 18. 3) Eq. (18.7) in Engel/Reid Good news: angular part (20.2) = (18.7)

35 Engel s book page 436 Why are the solutions for the rigid rotor and the angular components of the H atom identical?

36 Separation of variables 3D rigid rotor Radial equation

37 Radial equation repulsive effective potential attractive

38 Spherical harmonics 3D rigid rotor

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40 Radial functions

41 H atom wave functions PChem Quantum mechanics n l m ψ n l m

42 Real wave functions as linear combinations of complex wave functions

43 Quantum numbers for one electron system (H atom) PChem Quantum mechanics n = 1, 2, 3, l = 0, 1, 2, 3,, n-1 principal quantum number m l = 0, ±1, ± 2, ± 3,, ± l angular momentum quantum number magnetic quantum number

44 Paul Adrien Maurice Dirac ( ) was a British theoretical physicist. Dirac made fundamental contributions to the early development of both quantum mechanics and quantum electrodynamics. He held the Lucasian Chair of Mathematics at the University of Cambridge and spent the last fourteen years of his life at Florida State University. Among other discoveries, he formulated the Dirac equation, which describes the behaviour of fermions which led to the prediction of the existence of antimatter. Dirac shared the Nobel Prize in physics 1933 with Erwin Schrödinger, "for the discovery of new productive forms of atomic theory."

45 Stern-Gerlach experiment Ag: closed-shell + 5s electron No angular momentum Effect must be related to 5s electron l=0 for s-electrons It cannot be the angular momentum Electron spin

46 Electron spin vs. orbital angular momentum Orbital momentum L = l( l +1) l = 0,1,2, (n-1) S = s( s + 1) = Electron Spin 3 4 PChem Quantum mechanics s = 1/2 L z = m l m l = +l,,0, -l S z = m s m s = +1/2, -1/2 special case for l = 1

47 Degeneracy of states Two or more different quantum states are degenerate if they have the same energy. 0 Energy l = n n 3 l = m= n Allowed energies E = (degeneracy of states) 2 9 Number of states with identical energy n = 1 Rd 1

48 For a spherical problem use the Schrödinger Eq. in polar/spherical coordinates Use Coulomb potential V=-1/r Separation of variables nlm,. (,, ) = () Θ() Φ() ψ rφθ Rr θ φ Electron is confined to Coulomb potential, boundary condition, Quantum numbers n, l, m, s and rules for these E = -1/n 2, degeneracy of states Read this on a snowy weekend

49 Solving the Schrödinger equation. What do we get out of this? $ 1) wave functions probabilities, electron densities 2) e - confined to potential quantum numbers and rules how to use them 3) allowed energies interpretation of spectroscopic data

50 -) What is a Coulomb potential? -) Write the Hamiltonian for H atom in Cartesian coordinates. -) Outline the basic ideas of the solution of the H atoms in Q.M. -) Why are the solutions for the rigid rotor and the angular components of the H atom identical? -) In what functions does the wave function separate? -) What is a radial function? -) Why are linear combinations of the wave function used? -) What are the relevant quantum numbers? -) What are the energy eigenvalues? -) What is the degeneracy of states?

51 -) What is an emission spectrum? -) What is an absorption spectrum? -) Sketch qualitatively the emission pattern of a gas such as helium or hydrogen -) List Bohr s postulates -) What was the main accomplishment of Bohr s model? -) The Balmer series revers to good luck in a row when playing lottery? Yes/no. -) What is the Lyman series? -) What are stationary states in the framework of Bohr s model? -) Does an electron emit radiation in a stationary state? -) Only the energy is quantized but certainly not the angular momentum. Yes/no. -) The emission lines of Helium are evenly spaced in the frequency domain. Yes/no. -) Hydrogen and deuterium will result in the same emission spectra? Yes/no -) The Rydberg constant revers to the fact that the stock market always goes up over time, i.e. don t worry about your retirement as Rydberg told us already Yes/no -) Why is the reduced mass required? What is this?

52 Figure acknowledgement All images shown in this power point presentation were made by the author except the following with are excluded for the copyright of the author: xxx No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means except as permitted by the United States Copyright Act, without prior written permission of the author. Trademarks and copyrights are property of their respective owners., 2016 Publisher and author: Uwe Burghaus, Fargo, ND, USA